Modules with Finitely Many Submodules

S. Akbari; S. Khalashi Ghezelahmad; E. Yaraneri

doi:10.1142/S1005386716000444

Modules with Finitely Many Submodules

S. Akbari, S. Khalashi Ghezelahmad, E. Yaraneri

Department of Mathematics Engineering

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.

Original language	English
Pages (from-to)	463-468
Number of pages	6
Journal	Algebra Colloquium
Volume	23
Issue number	3
DOIs	https://doi.org/10.1142/S1005386716000444
Publication status	Published - 1 Sept 2016

Bibliographical note

Publisher Copyright:
© 2016 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.

Funding

The first author is indebted to School of Mathematics, Institute for Research in Funda-mental Sciences (IPM) for support. The research of the first author was supported in part by a grant from IPM (No. 93050212).

Funders	Funder number
Institute for Research in Funda-mental Sciences
Institute for Research in Fundamental Sciences

Keywords

direct sum
module
submodule

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10.1142/S1005386716000444

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AB - In this paper, we study modules having only finitely many submodules over any ring which is not necessarily commutative. We try to understand how such a module decomposes as a direct sum. We justify that any module V having only finitely many submodules over any ring A is an extension of a cyclic A-module by a finite A-module. Under some assumptions on A, such as commutativity of A, we prove that an A-module V has finitely many submodules if and only if V can be written as a direct sum of a cyclic A-module having only finitely many A-submodules and a finite A-module.

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ER -

Modules with Finitely Many Submodules

Abstract

Bibliographical note

Funding

Keywords

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